Convergence rate of strong Local Linearization schemes for stochastic differential equations with additive noise

There is a variety of strong Local Linearization (LL) schemes for the numerical integration of stochastic differential equations with additive noise, which differ with respect to the algorithm that is used in the numerical implementation of the strong Local Linear discretization. However, in contrast with the Local Linear discretization, the convergence rate of the LL schemes has not been studied so far. In this paper, two general theorems about this matter are presented and, with their support, additional results are derived for some particular schemes. As a direct application, the convergence rate of some strong LL schemes for SDEs with jumps is briefly expounded as well.     

Weak exponential schemes for stochastic differential equations with additive noise

^** Email: cmora{at}ing-mat.udec.cl This paper develops weak exponential schemes for the numericalsolution of stochastic differential equations (SDEs) with additivenoise. In particular, this work provides first and second-ordermethods which use at each iteration the product of the exponentialof the Jacobian of the drift term with a vector. The articlealso addresses the rate of convergence of the new schemes. Moreover,numerical experiments illustrate that the numerical methodsintroduced here are a good alternative to the standard integratorsfor the long time integration of SDEs whose solutions by thecommon explicit schemes exhibit instabilities.     

SEMI-LINEAR SYSTEMS OF BACKWARD STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN R~n

This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs. The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions.     

SEMI-LINEAR SYSTEMS OF BACKWARD STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN $R^n$

This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs.The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions.     

分数布朗单的幂函数随机积分逼近

本文研究了分数布朗单的逼近问题.利用Wiener积分,得到了分数布朗单的幂函数型随机积分逼近.     

RELATIONS BETWEEN SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY SEMIMARTINGALE WITH NON-LIPSCHITZ COEFFICIENTS

In this paper, a class of stochastic differential equations (SDEs) driven by semi-martingale with non-Lipschitz coefficients is studied. We investigate the dependence of solutions to SDEs on the initial value. To obtain a continuous version, we impose the conditions on the local characteristic of semimartingale. In this case, it gives rise to a flow of homeomorphisms if the local characteristic is compactly supported.     

关于随机积分的一点注记

本文给出随机积分的一种新的逼近方法．构造了一种统一而具体的构造程序，并利用这一程序解决了有关随机积分的分布和随机微分方程的变量代换的问题．     

随机扰动下统一混沌系统的有限时间同步

本文研究了具有随机扰动的统一混沌系统的有限时间同步问题,其中随机扰动是一维标准的维纳随机过程.利用了有限时间随机李雅普诺夫稳定性理论、伊藤公式,本文分三个步骤设立了三个控制器获得了驱动–响应系统在有限时间内的均方渐近同步.最后进行的数值模拟验证了理论结果的正确性和方法的有效性.     

High strong order stochastic Runge-Kutta methods for Stratonovich stochastic differential equations with scalar noise

This paper concerns the stochastic Runge-Kutta (SRK) methods with high strong order for solving the Stratonovich stochastic differential equations (SDEs) with scalar noise. Firstly, the new SRK methods with strong order 1.5 or 2.0 for the Stratonovich SDEs with scalar noise are constructed by applying colored rooted tree analysis and the theorem of order conditions for SRK methods proposed by Rößler (SIAM J. Numer. Anal. 48(3), 922–952, 2010). Secondly, a specific SRK method with strong order 2.0 for the Stratonovich SDEs whose drift term vanishes is proposed. And another specific SRK method with strong order 1.5 for the Stratonovich SDEs whose drift and diffusion terms satisfy the commutativity condition is proposed. The two specific SRK methods need only to use one random variable and do not need to simulate the multiple Stratonovich stochastic integrals. Finally, the numerical results show that performance of our methods is better than those of well-known SRK methods with strong order 1.0 or 1.5.     

A class of orthogonal integrators for stochastic differential equations

The purpose of this paper is to construct a class of orthogonal integrators for stochastic differential equations (SDEs). The family of SDEs with orthogonal solutions is univocally characterized. For this, a class of orthogonal integrators is introduced by imposing constraints to Runge–Kutta (RK) matrices and weights of the standard stochastic RK schemes.The performance of the method is illustrated by means of numerical simulations.     

CONVERGENCE ANALYSIS OF PARAREAL ALGORITHM BASED ON MILSTEIN SCHEME FOR STOCHASTIC DIFFERENTIAL EQUATIONS

In this paper, we propose a parareal algorithm for stochastic differential equations (SDEs), which proceeds as a two-level temporal parallelizable integrator with the Milstein scheme as the coarse propagator and the exact solution as the fine propagator. The convergence order of the proposed algorithm is analyzed under some regular assumptions. Finally, numerical experiments are dedicated to illustrating the convergence and the convergence order with respect to the iteration number $k$, which show the efficiency of the proposed method.     

Implicit Milstein method for stochastic differential equations via the Wong-Zakai approximation

The aim of this paper is to derive a numerical scheme for solving stochastic differential equations (SDEs) via Wong-Zakai approximation. One of the most important methods for solving SDEs is Milstein method, but this method is not so popular because the cost of simulating the double stochastic integrals is high. For overcoming this complexity, we present an implicit Milstein scheme based on Wong-Zakai approximation by approximating the Brownian motion with its truncated Haar expansion. The main advantages of this method lie in the fact that it preserves the convergence order and also stability region of the Milstein method while its simulation is much easier than Milstein scheme. We show the convergence rate of the method by some numerical examples.     

Asymptotic Mean-Square Stability of Two-Step Methods for Stochastic Ordinary Differential Equations

We deal with linear multi-step methods for SDEs and study when the numerical approximation shares asymptotic properties in the mean-square sense of the exact solution. As in deterministic numerical analysis we use a linear time-invariant test equation and perform a linear stability analysis. Standard approaches used either to analyse deterministic multi-step methods or stochastic one-step methods do not carry over to stochastic multi-step schemes. In order to obtain sufficient conditions for asymptotic mean-square stability of stochastic linear two-step-Maruyama methods we construct and apply Lyapunov-type functionals. In particular we study the asymptotic mean-square stability of stochastic counterparts of two-step Adams–Bashforth- and Adams–Moulton-methods, the Milne–Simpson method and the BDF method. AMS subject classification (2000) 60H35, 65C30, 65L06, 65L20     

A non-standard-Euler–Maruyama scheme

We construct a non-standard finite difference numerical scheme to approximate stochastic differential equations (SDEs) using the idea of weighed step introduced by R.E. Mickens. We prove the strong convergence of our scheme under locally Lipschitz conditions of a SDE and linear growth condition. We prove the preservation of domain invariance by our scheme under a minimal condition depending on a discretization parameter and unconditionally for the expectation of the approximate solution. The results are illustrated through the geometric Brownian motion. The new scheme shows a greater behaviour compared with the Euler–Maruyama scheme and balanced implicit methods which are widely used in the literature and applications.     

Numerical Solution to Hybrid Stochastic Differential Systems

Abstract

In this article numerical methods for solving hybrid stochastic differential systems of Itô-type are developed by piecewise application of numerical methods for SDEs. We prove a convergence result if the corresponding method for SDEs is numerically stable with uniform convergence in the mean square sense. The Euler and Runge–Kutta methods for hybrid stochastic differential equations are specifically described and the order of the error is given for the Euler method. A numerical example is given to illustrate the theory.     

On the practical global uniform asymptotic stability of stochastic differential equations

The method of Lyapunov functions is one of the most effective ones for the investigation of stability of dynamical systems, in particular, of stochastic differential systems. The main purpose of the paper is the analysis of the stability of stochastic differential equations (SDEs) by using Lyapunov functions when the origin is not necessarily an equilibrium point. The global uniform boundedness and the global practical uniform exponential stability of solutions of SDEs based on Lyapunov techniques are investigated. Furthermore, an example is given to illustrate the applicability of the main result.     

一种改进的随机次梯度方法的快速收敛性

1引言考虑如下优化问题: min f(x)=sum from i=1 to m f_i(x),s.t. x∈X (1)其中,f_i∶R~n→R是凸函数且f_i不可微,X是R~n上的非空闭凸子集．解(1)的主要方法     

带跳随机波动率模型的高阶ADI分裂格式

针对带跳随机波动率模型满足的偏积分微分方程,提出一种新的高阶交替方向隐式（ADI）有限差分格式,该模型是一个具有混合导数和非常数系数的对流扩散型初边值问题.我们将不同的高阶空间离散与时间步ADI分裂格式相结合,得到了一种空间四阶精度、时间二阶精度的有效方法,并采用Fourier方法分析了高阶ADI格式的稳定性.最后,通过对欧式看跌期权定价模型进行数值实验证实了数值方法的高阶收敛性.     

Numerical analysis of semilinear stochastic evolution equations in Banach spaces

The solution of stochastic evolution equations generally relies on numerical computation. Here, usually the main idea is to discretize the SPDE spatially obtaining a system of SDEs that can be solved by e.g., the Euler scheme. In this paper, we investigate the discretization error of semilinear stochastic evolution equations in L^p-spaces, resp. Banach spaces. The space discretization may be done by Galerkin approximation, for the time discretization we consider the implicit Euler, the explicit Euler scheme and the Crank–Nicholson scheme. In the last section, we give some examples, i.e., we consider an SPDEs driven by nuclear Wiener noise approximated by wavelets and delay equation approximated by finite differences.     

滞后型分段连续随机微分方程的稳定性

本文研究了滞后型分段连续随机微分方程的解析稳定性和数值稳定性问题.首先,利用伊藤公式等方法获得了解析解均方稳定的条件,其次,对于包括均方稳定和T-稳定在内的Euler-Maruyama方法的数值稳定性问题,运用不等式技术和随机分析方法获得了一些新的结果,证明了在一定条件下,Euler-Maruyama方法既是均方稳定又是T-稳定的,推广了随机延迟微分方程的数值稳定性结论.